Closed and anomalous domain walls

In Fig. 9, note that X_n is not the bulk of an−1 but uniquely determined by an−1. We define Z_n−1⁽¹⁾ (an−1) :=X_n. Note thatan−1can be viewed as a closed wall betweenA_nandX_n⊠_Zn(B_n)B_n or a closed wall between A^op_n ⊠_Zn(A_n)X_n and B_n. It is not a closed wall between A_n and B_n unless X_n is trivial.

A_n an−1 B_n bn−1 C_n cn−1 D_n

Zn(A) Zn(B) Zn(C) Zn(D)

X_n Y_n Z_n

Figure 9: The strong unique-bulk hypothesissays that, in above physical configuration, not only the (n+1)D bulks Zn(A), Zn(B), Zn(C) and Zn(D) are uniquely determined by A,B,C,D, respectively, but also the nD domain walls X_n is uniquely determined by (A, a,B), and Y_n by (B, b,C), andZ_n by (C, c,D). We also denoteX_n byZn⁽¹⁾(an−1).

Definition 6.1. A domain wallan−1between theA_n-phase and theB_n-phase is called Morita-closedifX_n is invertible; it is calledclosedifZn(A_n) =Zn(B_n)⁹ andX_n= idZn(A_n); it is called anomalousifX_n is not invertible.

Definition 6.2. If two simple topological ordersA_n andB_n are connected by a Morita-closed gapped domain wall an−1, then we say that they areMorita equivalent, denoted by A_n ∼B_n, and the Morita equivalence is given by an−1. When A_n and B_n are both closed, the Morita equivalence is also called theWitt equivalence[DMNO, FSV1, Ko2, KW].

By the definition of Morita equivalence, we have

A_n∼B_n ⇒ Zn(An)≃ Zn(Bn). (6.1)

This physical result is also natural mathematically. In mathematics, Morita equivalent algebras in a certain nice monoidal category always share the same center. Actually, it is also interesting to consider another direction. In general,Z(A)≃Z(B) does not implyA∼B. But if we assume certain duality structures on AandB, then it is possible. We give a couple of examples below.

1. In 1D, if Aand B are two finite dimensional C^∗-algebras, i.e. the direct sums of matrix algebras, describing composite (unstable) topological orders, thenA is Morita equivalent to B if and only if Z(A)≃ Z(B) as algebras. This result is quite trivial. But it can be generalized to a non-trivial one in the framework of 2D rational conformal field theory, in whichAandBare two special symmetric Frobenius algebras in a modular tensor categories, thenA∼B if and only ifZ(A)≃Z(B) as algebras, whereZ(A) is the so-calledfull center ofA[KR1].

2. In 2D, if two unitary fusion 1-categoriesA₂ and B₂ are Morita equivalent if and only if their Drinfeld centers are equivalent as braided monoidal 1-categories [M1, Ki3, ENO08].

It seems natural to ask if the Morita equivalence is equivalent to the equivalence of Zn-centers for a unitary fusionn-category. It turns out that it is not true. In 3D, all closed 3D topological orders share the same trivial bulk, but they are not Morita equivalent (or Witt equivalent) in general. The discrepancy is measured by the Witt equivalence classes, which form an infinite group called Witt group [DMNO]. What happens is that, assumingZn(An) =Zn(Bn), one can certainly glue the topological phaseZn(An) (with a gapped boundaryA_n) with the phaseZn(Bn) (with a gapped boundary B_n) smoothly in the n+1Dbulk and create an n−1D wall between A_n and B_n on the boundary. But this n−1D wall can be gapless in general. For example, a 3D quantum hall system shares the same 4D bulkwith 13, but the domain wall between them

9When we sayZ_n(An) =Z_n(Bn), we mean that we have made a choice of how we identify them.

Figure 10: A Morita-closed wall between two smooth boundaries in the toric code model is gapless. In general, the Witt classes of closed nD topological orders also form a group, still called Witt group [KW], which measures the discrepancy between the Witt equivalence and the equivalence of their bulk’s.

Example 6.3. We give an example of a Morita-closed wall in the toric code model. Consider the toric code model with a smooth boundary, a 1-codimensional wall and a 2-codimensional defect on the boundary depicted in Fig. 10. This model is completely free of frustration. The complete list of mutually commutative stabilizers are given as follows:

Av =σ^x₁σ₂^xσ³σ⁴, Bp=σ^z₈σ^z₁₀σ^z₁₁σ₁₂^z , A13,14,15=σ^x₁₃σ^x₁₄σ₁₅^x

C2,5,3|7=σ₂^zσ₅^zσ₃^zσ^x₇, D3|7,8,9=σ^x₃σ₇^zσ₈^zσ₉^z, Q6,17,18,19,20=σ₆^xσ^y₁₇σ^z₁₈σ₁₉^z σ₂₀^z .

The excitations on the smooth boundary are given by the unitary fusion 1-categoryRepZ₂ [BK, KK]. Since the dotted line is an invertible wall that gives the EM-duality [KK], the neighborhood of the plaquette (17,18,19,20) can be viewed as a 0+1D Morita-closed wall between two smooth boundaries. Note that moving an e/m-particle around the corner of the edges labeled by 6 and 17 (the blue dot) turn it into an m/e-particle. Since an m-particle condenses on the smooth boundary, all particles condense in the neighborhood of the plaquette (17,18,19,20). Therefore, the Morita-closed wall on the boundary contains no non-trivial excitations, thus can be described by1₁ =Hilb. Indeed, Hilb is at the same time an invertibleRepZ₂-bimodule and gives a non-trivial Morita equivalence between two smooth boundariesRepZ₂. This Morita equivalenceHilb between RepZ₂ and RepZ₂ determines exactly an invertible domain wall (the dotted line) with wall excitations given byFunRep_Z2|Rep_Z2(Hilb,Hilb), which is also equivalent to the EM-duality of the bulk, i.e. a braided auto-equivalence ofZ2(RepZ₂).

Example 6.4. We give some examples of Morita-closed/anomalous walls in Levin-Wen type of models enriched by boundaries and defects depicted in Fig. 11. LetA2=B2be unitary fusion 1-categories andZ₂⁽⁰⁾(A₂) :=Z2(A₂). Fig. 11 depicts a lattice model with a gappedM-wall between Z2(A2) and Z2(B2). The wall excitations are given by unitary fusion 1-categoryZ₂⁽¹⁾((M, a)), which is defined as follows:

Z₂⁽¹⁾((M, a)) =Z₂⁽¹⁾(M) :=FunA|B(M,M)^rev. (6.2) An A-module functor a∈FunA(A,M⊠BB)≃M gives a defect junction betweenA₂, B₂ and Z₂⁽¹⁾(M). If M is invertible, then a gives a Morita-closed domain wall between A₂ and B₂; if

A₂ a B₂ Z2(A) Z2(B)

Z₂⁽¹⁾(M)

Figure 11: Consider a Levin-Wen model enriched by gapped boundaries and defects [KK]. The two bulk lattices are defined by unitary fusion 1-categories A2 and B2, respectively. The exci-tations in the left/right bulk are given by the Drinfeld center Z2(A2)/Z2(B2) of A2/B2. The lattice near the wall between the left bulk and the right bulk is defined by a semisimple inde-composable A-B-bimodule M₁, and is called the M-wall. The excitations on the M-wall are given by the unitary fusion 1-category Z₂⁽¹⁾(M₁) := FunA|B(M,M)^rev [KK]. Two boundary lattices are defined by Aand B, viewed as a left A-module and a left B-module, respectively, and are called the A-boundary and the B-boundary. The excitations on the A-boundary are given by A ≃ Fun^A(A,A); those on the B-boundary are given by B. The defect junction connecting the A-boundary, M-wall and B-boundary is given by a unitary C-module functor a∈Fun^A(A,M⊠BB)≃M.

otherwise, a is an anomalous domain wall. Whena is viewed as an 1D topological order, it is nothing but (M, a). This example shows that, all semisimple indecomposable A-B-bimodules are physically realizable as (potentially anomalous) 1D domain walls in Levin-Wen models. Also note that if we fold two boundaries in Fig. 11 upward, we obtain a vertical line with the bottom end given by the 1D topological order (M, a). The vertical line is a 2D topological order given by a unitary multi-fusion 1-categoryFun(M,M), which is nothing but the Z₁⁽⁰⁾-center of (M₁, a).

Remark 6.5. We believe that higher dimensional generalization of Levin-Wen models can be constructed by replacing unitary fusion 1-categories by unitary fusionn-categories (see a special case in [WWa] and a sketch of general construction in [Wa]). We believe that a large part of Example 6.4 (Fig. 11) remains to be true forn >2.

LetA_n andB_n be two simplenD topological orders. Note thatZn⁽¹⁾(−) defines a surjective map from potentially anomalous (or Morita-closed) domain walls betweenA_n andB_n to closed (or invertible) domain walls between Zn(An) and Zn(Bn). Sometimes, a stronger result can be obtained. When n= 2,A₂ and B₂ are unitary fusion 1-categories. In this case, Z₂⁽¹⁾ maps bijectively from (potentially anomalous) walls betweenA₂andB₂to closed walls betweenZ2(A2) and Z2(B2) [DMNO, KZ] (see also Thm. 6.10). Moreover, Morita-closed domain walls between A₂ and B₂ are nothing but invertible A₂-B₂-bimodules. When A₂ = B₂, they form a group denoted by Pic(A₂). We have the following group isomorphism [ENO09, KK]:

Z₂⁽¹⁾: Pic(A2)−^≃→Aut(Z2(A2)), (6.3) where Aut(Z2(A₂)) is the group of invertible walls between Z2(A₂) and itself, or equivalently, the group of braided auto-equivalences ofZ2(A₂).

Remark 6.6. The isomorphism in Eq. (6.3) is not an isolated phenomenon. In 1D, if A₁ is a simple algebra overC, this result is trivial. In 2D rational conformal field theories, what replaces A₂is a simple special symmetry Frobenius algebraAin a modular tensor category. In this case, we also have a group isomorphism Pic(A)≃Aut(Z(A)) [DKR1], where Z(A) is the full center of A, with a similar physical meaning [FrFRS]. In the framework of 3D TQFT’s, the similar phenomenon was studied recently in [FS, FPSV].